On the disturbance of the steady flow of an inviscid liquid between parallel planes

1934 ◽  
Vol 234 (732) ◽  
pp. 43-78 ◽  
Keyword(s):  
Mathematical Proof ◽  
Fluid Motion ◽  
Steady Motion ◽  
Circular Cylinders ◽  
General Validity ◽  
Inviscid Liquid ◽  
Characteristic Value ◽  
Characteristic Values ◽  
The Stability ◽  
Liquid Flows

1—In a number of papers dealing with the stability of fluid motion, RAYLEIGH employed a certain method, which we may refer to as the “characteristic-value” method. For some problems this method gives results in agreement with observation. For example, it establishes that a heterogeneous inviscid liquid at rest under gravity is stable if the density decreases steadily as we pass upward; it establishes that an inviscid liquid rotating between concentric circular cylinders is stable if, and only if, the square of the circulation increases steadily as we pass outward. This result was stated by RAYLEIGH, and its validity appears to be confirmed by the experiments of TAYLOR, but a simple < d 2 u 0 /dy 2 retains the same sign throughout the liquid, u 0 being the velocity in the steady motion and y the distance from one of the planes. This result is deduced from the fact that mathematical proof by the characteristic value method was not given. I have recently supplied such a proof, extending the problem to include a heterogeneous liquid. But when the method is applied to some other problems, the situation is not so satisfactory. Among the results to which Rayleigh was led is the following. If an inviscid liquid flows between parallel planes, the motion is stable if the characteristic values of a parameter in a certain differential equation cannot be complex, the implication being that they are therefore real. Rayleigh further claimed that the method established the stability of a uniform shearing motion, for which d 2 u 0 /dy 2 =0. KELVIN and LOVE criticized the method, and a review of the situation in 1907 was given by ORR. In spite of the fact that its general validity remains obscure, the characteristic-value method has been widely employed. It is not the purpose of the present paper to attempt to justify or to discredit the characteristic-value method in general. The paper deals only with the simplest of all stability problems, that of an inviscid liquid flowing between fixed parallel planes. In §2 the method is discussed in some detail and in §3 an argument is developed to show that Rayleigh’s criterion for stability, mentioned above, cannot be legitimately deduced by his method. He proved that complex characteristic values are impossible, and I now prove that real characteristic values are also impossible. The conclusion to be drawn is that the characteristic-value method is not applicable to this case.

The stability of viscous fluid flow between rotating cylinders

1937 ◽  
Vol 33 (1) ◽  
pp. 41-61 ◽  
Author(s):  
S. Goldstein
Keyword(s):  
Angular Velocity ◽  
Wave Length ◽  
Steady Motion ◽  
Circular Cylinders ◽  
Critical Flow ◽  
Average Value ◽  
Time Period ◽  
Component Velocity ◽  
The Stability ◽  
Image Position

The stability of the motion of viscous incompressible fluid, of density ρ and kinematic viscosity ν, between two infinitely long coaxial circular cylinders, of radiiaanda+d, whered/ais small, is investigated mathematically by the method of small oscillations. The inner cylinder is rotating with angular velocity ω and the outer one with angular velocity αω, and there is a constant pressure gradient parallel to the axis. The fluid therefore has a component velocityWparallel to the axis, in addition to the velocity round the axis. A disturbance is assumed which is symmetrical about the axis and periodic along it. The critical disturbance, which neither increases nor decreases with the time, is periodic with respect to the time (except whenW= 0, when the critical disturbance is a steady motion). As Reynolds number of the flow we take ||d/ν, whereis the average value ofWacross the annulus, and we denote bylthe wave-length of the disturbance along the axis, by σ/2π the time period of the critical flow, bycthe wavelength of the critical flow, byωcthe critical value of ω, and we putapproximately, if α is not nearly equal to 1.

scholarly journals 1. Stability of Fluid Motion.—Rectilineal Motion of Viscous Fluid between two Parallel Planes

1888 ◽  
Vol 14 ◽  
pp. 359-368 ◽  
Author(s):  
W. Thomson
Keyword(s):  
Viscous Fluid ◽  
Fluid Motion ◽  
Steady Motion ◽  
Philosophical Magazine ◽  
University Of Cambridge ◽  
The Stability ◽  
The University

Since the communication of the first of this series of articles to the Eoyal Society of Edinburgh in April, and its publication in the Philosophical Magazine in May and June, the stability or instability of the steady motion of a viscous fluid has been proposed as subject for the Adams Prize of the University of Cambridge for 1888.

Development and Effects of Super-Critical Taylor-Vortex Flow in a Lightly Loaded Journal Bearing

1974 ◽  
Vol 96 (1) ◽  
pp. 28-35 ◽  
Author(s):  
R. C. DiPrima ◽  
J. T. Stuart
Keyword(s):  
Vortex Flow ◽  
Axial Direction ◽  
Basic Flow ◽  
Circular Cylinders ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Secondary Motion ◽  
The Stability ◽  
Unified Description

At sufficiently high operating speeds in lightly loaded journal bearings the basic laminar flow will be unstable. The instability leads to a new steady secondary motion of ring vortices around the cylinders with a regular periodicity in the axial direction and a strength that depends on the azimuthial position (Taylor vortices). Very recently published work on the basic flow and the stability of the basic flow between eccentric circular cylinders with the inner cylinder rotating is summarized so as to provide a unified description. A procedure for calculating the Taylor-vortex flow is developed, a comparison with observed properties of the flow field is made, and formulas for the load and torque are given.

scholarly journals Stability analysis of the coexistence equilibrium of a balanced metapopulation model

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Shodhan Rao ◽  
Nathan Muyinda ◽  
Bernard De Baets
Keyword(s):  
Equilibrium Point ◽  
Mathematical Proof ◽  
Mean Field ◽  
Patch Dynamics ◽  
System Modelling ◽  
Metapopulation Model ◽  
Ode System ◽  
Homogeneous Network ◽  
The Stability ◽  
Coexistence Equilibrium

AbstractWe analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, more specifically, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration. We assume that the inter-patch migrations are detailed balanced and that the patches are identical with intra-patch dynamics governed by a mean-field ODE system with a coexistence equilibrium. By making use of an appropriate Lyapunov function coupled with LaSalle’s invariance principle, we are able to show that the coexistence equilibrium point within each patch is locally asymptotically stable if the inter-patch dispersal network is heterogeneous, whereas it is neutrally stable in the case of a homogeneous network. These results provide a mathematical proof confirming the existing numerical simulations and broaden the range of networks for which they are valid.

scholarly journals Effect of Rotational Speed on the Stability of Two Rotating Side-by-side Circular Cylinders at Low Reynolds Number

2018 ◽  
Vol 27 (2) ◽  
pp. 125-134
Author(s):  
Huashu Dou ◽  
Shuo Zhang ◽  
Hui Yang ◽  
Toshiaki Setoguchi ◽  
Yoichi Kinoue
Keyword(s):  
Reynolds Number ◽  
Rotational Speed ◽  
Low Reynolds Number ◽  
Circular Cylinders ◽  
Low Reynolds ◽  
The Stability

Experiments on the stability of sinusoidal flow over a circular cylinder

2002 ◽  
Vol 457 ◽  
pp. 157-180 ◽  
Author(s):  
TURGUT SARPKAYA
Keyword(s):  
Stokes Flow ◽  
Coherent Structures ◽  
High Speed ◽  
Three Dimensional ◽  
Separated Flow ◽  
Circular Cylinders ◽  
Centrifugal Forces ◽  
Complex Interactions ◽  
Sinusoidal Flow ◽  
The Stability

The instabilities in a sinusoidally oscillating non-separated flow over smooth circular cylinders in the range of Keulegan–Carpenter numbers, K, from about 0.02 to 1 and Stokes numbers, β, from about 103 to 1.4 × 106 have been observed from inception to chaos using several high-speed imagers and laser-induced fluorescence. The instabilities ranged from small quasi-coherent structures, as in Stokes flow over a flat wall (Sarpkaya 1993), to three-dimensional spanwise perturbations because of the centrifugal forces induced by the curvature of the boundary layer (Taylor–Görtler instability). These gave rise to streamwise-oriented counter-rotating vortices or mushroom-shaped coherent structures as K approached the Kh values theoretically predicted by Hall (1984). Further increases in K for a given β led first to complex interactions between the coherent structures and then to chaotic motion. The mapping of the observations led to the delineation of four states of flow in the (K, β)-plane: stable, marginal, unstable, and chaotic.

Non-local effects in the stability of flow between eccentric rotating cylinders

1972 ◽  
Vol 54 (3) ◽  
pp. 393-415 ◽  
Author(s):  
R. C. Diprima ◽  
J. T. Stuart
Keyword(s):  
Differential Equations ◽  
Stability Theory ◽  
Taylor Number ◽  
Circular Cylinders ◽  
Taylor Vortex ◽  
Local Theory ◽  
Local Instability ◽  
Clearance Ratio ◽  
Linearized Stability ◽  
The Stability

In this paper the linear stability of the flow between two long eccentric rotating circular cylinders is considered. The problem, which is of interest in lubrication technology, is an extension of the classical Taylor problem for concentric cylinders. The basic flow has components in the radial and azimuthal directions and depends on both of these co-ordinates. As a consequence the linearized stability equations arepartial differential equationsrather than ordinary differential equations. Thus standard methods of stability theory are not immediately useful. However, there are two small parameters in the problem, namely δ, the clearance ratio, and ε, the eccentricity. By letting these parameters tend to zero in such a way that δ½ is proportional to ε, a global solution to the stability problem is obtained without recourse to the concept of ‘local instability’, or ‘parallel-flow’ approximation, so commonly used in boundary-layer stability theory. It is found that the predictions of the present theory are at variance with what is given by a ‘local’ theory. First, the Taylor-vortex amplitude is found to be largest at about 90° downstream of the region of ‘maximum local instability’. This result is given support by some experimental observations of Vohr (1968) with δ = 0·1 and ε = 0·475, which yield a corresponding angle of about 50°. Second, the critical Taylor number rises with ε, rather than initially decreasing with ε as predicted by local stability theory using the criteria of maximum local instability. The present prediction of the critical Taylor number agrees well with Vohr's experiments for ε up to about 0·5 when δ = 0·01 and for ε up to about 0·2 when δ = 0.1.

Stability of Horizontal Boundary Layers

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Mathematical Proof ◽  
Vertical Component ◽  
Ekman Layer ◽  
Numerical Results ◽  
Energy Estimates ◽  
Navier Stokes ◽  
The 1960S ◽  
The Stability

Let us now detail the stability properties of an Ekman layer introduced in Part I, page 11. First we will recall how to compute the critical Reynolds number. Then we will describe briefly what happens at larger Reynolds numbers. The first step in the study of the stability of the Ekman layer is to consider the linear stability of a pure Ekman spiral of the form where U∞ is the velocity away from the layer and ζ is the rescaled vertical component ζ = x3/√εν. The corresponding Reynolds number is Let us consider the Navier–Stokes–Coriolis equations, linearized around uE The problem is now to study the (linear) stability of the 0 solution of the system (LNSCε). If u=0 is stable we say that uE is linearly stable, if not we say that it is linearly unstable. Numerical results show that u=0 is stable if and only if Re<Rec where Rec can be evaluated numerically. Up to now there is no mathematical proof of this fact, and it is only possible to prove that 0 is linearly stable for Re<Re1 and unstable for Re>Re2 with Re1<Rec<Re2, Re1 being obtained by energy estimates and Re2 by a perturbative analysis of the case Re=∞. We would like to emphasize that the numerical results are very reliable and can be considered as definitive results, since as we will see below, the stability analysis can be reduced to the study of a system of ordinary differential equations posed on the half-space, with boundary conditions on both ends, a system which can be studied arbitrarily precisely, even on desktop computers (first computations were done in the 1960s by Lilly).

Fluid Statics and Buoyancy

1997 ◽  
Author(s):  
David Jon Furbish
Keyword(s):  
Coordinate System ◽  
Fluid Motion ◽  
Steady Motion ◽  
Linear Acceleration ◽  
Vertical Position ◽  
Global Coordinate System ◽  
Fluid Density ◽  
Vertical Coordinate ◽  
Isothermal Fluid ◽  
Static Fluid

Fluid statics concerns the behavior of fluids that possess no linear acceleration within a global (Earth) coordinate system. This includes fluids at rest as well as fluids possessing steady motion such that no net forces exist. Such motions may include steady linear motion within the global coordinate system as well as rotation with constant angular velocity about a fixed vertical axis. In this latter case, centrifugal forces must be balanced by centripetal forces (which arise, for example, from a pressure gradient acting toward the axis of rotation). Moreover, we assert that no relative motion between adjacent fluid elements exists. Fluid motion, if present, is therefore like that of a rigid body. In addition, we neglect molecular motions that lead to mass transport by diffusion. Thus, the idea of a static fluid is a macroscopic one. The developments in this chapter clarify how pressure varies with coordinate position in a static fluid. Both compressible and incompressible fluids are treated. In the simplest case in which the density of a fluid is constant, we will see that pressure varies linearly with vertical position in the fluid according to the hydrostatic equation. In addition, we will consider the possibility that fluid density is not constant. Then, variations in density must be taken into account when computing the pressure at a given position in a fluid column; the pressure arising from the weight of the overlying fluid no longer varies linearly with depth. In the case of an isothermal fluid, whose temperature is constant throughout, any variation in density must arise purely from the compressible behavior of the fluid in response to variations in pressure. In the case where temperature varies with position, fluid density may vary with both pressure and temperature. We will in this regard consider the case of a thermally stratified fluid whose temperature varies only with the vertical coordinate direction. Because fluid statics requires treating how fluid temperature, pressure, and density are related, the developments below make use of thermodynamical principles developed in Chapter 4.

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